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Physical Review A

Geometric phase for a driven quantum field subject to decoherence

Shi‐Biao Zheng

Published May 22, 2015• 11 citations
Publisher

Abstract

We investigate the geometric phase for a quantum field driven by an external source and subjected to spontaneous decay from coupling to the reservoir. Starting from a coherent state, we analyze the effect of decoherence when the system undergoes a nonadiabatic cyclic evolution in phase space, and show that the lowest correction to the conventional geometric phase is quadratic in the decaying rate of the quantum field. This is in distinct contrast with the nonadiabatic geometric phase associated with the evolution of a spin system subjected to spontaneous decay, which contains the first-order correction term. We further show that the unconventional geometric phase is also robust to field decay.

DOI
10.1103/physreva.91.052117

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We investigate the geometric phase for a quantum field driven by an external source and subjected to spontaneous decay from coupling to the reservoir. Starting from a coherent state, we analyze the effect of decoherence when the system undergoes a nonadiabatic cyclic evolution in phase space, and show that the lowest correction to the conventional geometric phase is quadratic in the decaying rate of the quantum field. This is in distinct contrast with the nonadiabatic geometric phase associated with the evolution of a spin system subjected to spontaneous decay, which contains the first-order correction term. We further show that the unconventional geometric phase is also robust to field decay.
We investigate the behaviour of the geometric phase of a bipartite system with intersubsystem coupling via a quantum jump approach, where one of the subsystems is driven by a one- or two-mode quantized field which is also subjected to decoherence. The results show that the first correction due to the decohering field is only quadratic in decaying rate λ when the evolution of the system is adiabatic, and the lowest order correction is related to the first order in λ when the evolution of the system is non-adiabatic. Moreover, an interesting interpretation of this phenomenon is given.
Quantum computer takes advantages from properties of superposition and entanglement, which are also very fragile and easy to be error. But they may be destroyed easily by a process called decoherence. Therefore, how to overcome the effect of decoherence is key to implement quantum computer. Using the no-jump trajectory, Berry's and AA's formula for geometric phase, we calculate the geometric phase of a spin-1/2 system driven by one and two mode quantum fields subject to decoherence in the adiabatic and nonadiabatic case, respectively. By quantum jump method, we find that the corrections to its phase for the no-jump trajectory are different when considering adiabatic and nonadiabatic evolutions. Finally, we discuss the implications of our results from fundamental and quantum computational points.
We present a feasible scheme to investigate the geometric phase for an atom trapped in an optical cavity induced by the effective decay process due to cavity photon loss. The cavity mode, together with the external driving fields, acts as the engineered environment of the atom. When the parameters of the reservoir is adiabatically and cyclically changed, the system initially in the nontrivial dark state of the effective Lindblad operator undergoes a cyclic evolution and acquires a geometric phase. The geometric phase can be observed with the atomic Ramsey interference oscillation in the decoherence-free subspace.
Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing nonunitary, noncyclic, and nonadiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realization of quantum gates due to its robustness against dynamical errors; however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a nonperturbative control scheme.